Superfast second-order methods for unconstrained convex optimization

Yurii, Nesterov ()
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Yurii, Nesterov: Université catholique de Louvain, CORE, Belgium

No 2020007, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: In this paper, we present new second-order methods with converge rate O(k^{-4}), where k is the iteration counter. This is faster that athe existing lower bound for this type of schemes [1,2], which is O(k^{-7/2}). Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from [15] using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relati e non-degeneracy condition [3, 10]. During this process, the Hessian of the objective function is computed once, and the gradient is computed O(ln 1/epsilon) times, where epsilon is the desired accuracy of the solution for our problem.

Keywords: convex optimization; tensor methods; lower complexity bounds; second-order methods (search for similar items in EconPapers)
Date: 2020-02-10
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